永久式电路的有效除法下界 $n^2\log\log n$
An $n^2\log\log n$ Lower Bound for Permanent Circuits with Valid Division
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中文总结 AI 辅助
本文通过改进临界轨迹方法，结合列删除递推、最大秩定理和形式形变，证明了特征零域上永久式有理算术电路在有效除法下的 $n^2\log\log n$ 下界。
中文摘要 AI 辅助
我们证明了在特征为零的域上计算永久式的有理算术电路具有 $n^2\log\log n$ 的下界。加法和标量运算是免费的，而每次非标量乘法或有效除法都计为单位成本。若 $L_{\mathrm{div}}(\mathrm{per}_n)$ 表示由此得到的复杂度，则 $\liminf_{n\to\infty} L_{\mathrm{div}}(\mathrm{per}_n)/(n^2\log_2\log_2 n)\ge 1/12$。该证明以两种方式改进了永久式的临界轨迹方法。首先，列删除递推和包含矩阵的最大秩定理压缩了匹配-次式多项式临界方程中的共享系数空间。其次，依赖于电路的正式形变将所得的有限梯度切片通过任意有效除法电路转移。有限平坦性保持特殊纤维的长度，而范数论证使每个除数在一般形式纤维上成为单位。有理 Baur--Strassen 微分和仿射 Bézout 定理随后给出了具有所述常数的下界。
英文摘要
We record lower bounds for permanent circuits with valid division over characteristic zero, counting nonscalar multiplications and divisions while additions and scalar operations are free. Chapter 5 of OpenAI's Ten Advances supplies the block construction and critical-locus estimate; these, with the parameter choice made here and the classical bounds of Strassen and Baur-Strassen, give liminf as n tends to infinity of L_div(per_n)/(n^2 log_2 log_2 n) >= 1/12. Our finite-parameter refinement for matching-minor polynomials observes that the coefficient vector at deletion size h lies in a space of dimension at most min{binom(t,h), binom(t,d-h)}. Twelve machine-checked declarations of the Lean development accompanying OpenAI's manuscript on border determinantal complexity of the permanent (September 24, 2026) imply, by a short written argument, the existence of a geometric slice; a further written, unformalized application of the same criterion gives L_div(per_n) >= (n^2/119790) log_2(n/44) for n >= 1936. Assuming Proposition 7.3 of that manuscript gives the bound (n^2/86400) log_2(n/48) for n >= 1408. These are order-of-growth results; the elementary Hessian bound n^2/2 is larger at practical orders.
